Practical and effective methods of simulation based parameter estimation for multidimensional data
Schwalb, Otto W., III
Thompson, James R.
Doctor of Philosophy
In 1983, Atkinson, Bartoszynski, Brown, and Thompson proposed a method of parameter estimation referred to as "simulation based estimation", or SIMEST. SIMEST is closely related to maximum likelihood, in that both methods deal with parameter estimation in the context of a fully parametric model. With SIMEST, however, the arduous step of obtaining the density function from a set of model axioms is avoided via simulation. In this dissertation, we extend the ideas of SIMEST to the case of multidimensional data. A nearest neighbor based binning scheme is proposed where the observations are divided into bins determined by the "rings" of concentric ellipsoids, the "rings" being chosen to roughly approximate regions of equal probability. The ellipsoids are each allowed to have different axes, the axes for each ellipsoid being determined by the data. Some theoretical justification is developed which establishes strong consistency for the parameter estimates obtained by this method. The theory also suggests a promising variation on the idea using many overlapping bins. Another theoretical topic related to the problems associated with global optimization in SIMEST is also treated. We explore the usefulness of these techniques in modeling the secondary tumor generation mechanisms of cancer. In one model, it is assumed that 3-dimensional data is available: (a) the time from the detection and removal of the primary to the discovery of the first secondary tumor, (b) the volume of the primary tumor, and (c) the volume of the first secondary tumor. In a second model, it is assumed that two additional dimensions of information are available (i.e. 5-dimensional data): (d) the time from the detection of the first secondary tumor to the detection of the second secondary tumor and (e) the volume of the second secondary tumor.
Mathematics; Statistics; Oncology